Use "trichotomy". For any two real numbers, x and y, one and only one of these must apply:
1) x= y
2) x< y
3) y< x.

If [itex]d_1(a, b)< d_1(c, d)[/itex] then it is NOT possible that [itex]d_1(a,b)= d_1(c, d)[/itex] nor that [itex]d_1(c, d)> d_1(a, b)[/itex] which implies the same for [itex]d_2(a, b)[/itex] and [itex]d_2(c, d)[/itex].